Two-dimensional singular-value decomposition

In linear algebra, two-dimensional singular-value decomposition (2DSVD) computes the low-rank approximation of a set of matrices such as 2D images or weather maps in a manner almost identical to SVD (singular-value decomposition) which computes the low-rank approximation of a single matrix (or a set of 1D vectors).

SVD
Let matrix $$ X = [\mathbf x_1, \ldots, \mathbf x_n] $$ contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix $$ F $$ and Gram matrix $$ G $$
 * $$ F = X X^\mathsf{T} $$, $$ G = X^\mathsf{T} X, $$

and compute their eigenvectors $$ U = [\mathbf u_1, \ldots, \mathbf u_n] $$ and $$ V = [\mathbf v_1, \ldots, \mathbf v_n] $$. Since $$ VV^\mathsf{T} = I $$ and $$ UU^\mathsf{T} = I $$ we have
 * $$ X = UU^\mathsf{T} X VV^\mathsf{T} = U \left(U^\mathsf{T} XV\right) V^\mathsf{T} = U \Sigma V^\mathsf{T}. $$

If we retain only $$ K $$ principal eigenvectors in $$ U, V$$, this gives low-rank approximation of $$ X $$.

2DSVD
Here we deal with a set of 2D matrices $$ (X_1,\ldots,X_n) $$. Suppose they are centered $ \sum_i X_i =0 $. We construct row–row and column–column covariance matrices


 * $$ F = \sum_i X_i X_i^\mathsf{T} $$ and $$ G = \sum_i X_i^\mathsf{T} X_i $$

in exactly the same manner as in SVD, and compute their eigenvectors $$ U $$ and $$ V$$. We approximate $$ X_i $$ as


 * $$ X_i = U U^\mathsf{T} X_i V V^\mathsf{T} = U \left(U^\mathsf{T} X_i V\right) V^\mathsf{T} = U M_i V^\mathsf{T} $$

in identical fashion as in SVD. This gives a near optimal low-rank approximation of $$ (X_1,\ldots,X_n) $$ with the objective function


 * $$ J= \sum_{i=1}^n \left| X_i - L M_i R^\mathsf{T}\right| ^2 $$

Error bounds similar to Eckard–Young theorem also exist.

2DSVD is mostly used in image compression and representation.